Parameter Optimization

Hello everyone

For my bachelor thesis I am currently facing the following problem, it would be great if you could give me a hint how to approach it.

I have sample data consisting of pairs of numbers (A, B). There is an x% chance that the value of B is different to A. If it is different, value B is drawn as a random value from a certain distribution function F(x). For each pair of numbers I know, whether I know each value for certain or not. If I don't know a value for certain, it is drawn as a random value from the Poisson distribution, where the mean equals the true value.
Given this information I want to find x.

My approach so far was to, for each pair of numbers, calculate the probability that this configuration, given a certain x, occurs. For the case where I know A and B for certain, this would be:
if(A = B): (1-x)*F(x)
if(A <> B): x
Then I found out that I could get quite a good estimate of x, by maximizing the sum of the logarithms of the result for each pair of numbers.

This doesn't however work for the case where I don't know the two values for certain. Also I don't quite understand yet, why this maximization of the sum of logarithms seems to work.

For the case where I don't know A and B for certain I tried (without success):
(1-x)*Poisson(A,B) + x*F(x)

Thank you very much in advance for your help.

Hello, I just wonder, is there any information missing?